LMFDB - Embedded newform 378.5.b.a.323.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,5,Mod(323,378)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("378.323");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	378.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$39.0738460457$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.5443747577856.29
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 24x^{6} + 180x^{4} + 488x^{2} + 324 $ x^8 + 24x^6 + 180x^4 + 488*x^2 + 324
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{4}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			323.6
Root			$3.56118i$ of defining polynomial
Character	$\chi$	$=$	378.323
Dual form			378.5.b.a.323.3

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843i q^{2} -8.00000 q^{4} +3.12468i q^{5} +18.5203 q^{7} -22.6274i q^{8} +O(q^{10})$	\(q+2.82843i q^{2} -8.00000 q^{4} +3.12468i q^{5} +18.5203 q^{7} -22.6274i q^{8} -8.83793 q^{10} -141.266i q^{11} -237.623 q^{13} +52.3832i q^{14} +64.0000 q^{16} +394.261i q^{17} +535.891 q^{19} -24.9974i q^{20} +399.561 q^{22} +487.858i q^{23} +615.236 q^{25} -672.101i q^{26} -148.162 q^{28} -1201.77i q^{29} +296.662 q^{31} +181.019i q^{32} -1115.14 q^{34} +57.8699i q^{35} -380.977 q^{37} +1515.73i q^{38} +70.7034 q^{40} -73.6638i q^{41} +2912.26 q^{43} +1130.13i q^{44} -1379.87 q^{46} +2845.13i q^{47} +343.000 q^{49} +1740.15i q^{50} +1900.99 q^{52} +2125.45i q^{53} +441.411 q^{55} -419.066i q^{56} +3399.11 q^{58} +5486.16i q^{59} -5032.38 q^{61} +839.086i q^{62} -512.000 q^{64} -742.497i q^{65} +7342.60 q^{67} -3154.09i q^{68} -163.681 q^{70} +8600.64i q^{71} +2165.28 q^{73} -1077.57i q^{74} -4287.13 q^{76} -2616.28i q^{77} -5796.36 q^{79} +199.979i q^{80} +208.353 q^{82} +2410.24i q^{83} -1231.94 q^{85} +8237.10i q^{86} -3196.49 q^{88} +13791.5i q^{89} -4400.85 q^{91} -3902.87i q^{92} -8047.25 q^{94} +1674.49i q^{95} +10613.0 q^{97} +970.151i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 64 q^{4}+O(q^{10}) $ 8 * q - 64 * q^4	$ 8 q - 64 q^{4} - 224 q^{10} - 376 q^{13} + 512 q^{16} + 1120 q^{19} - 1120 q^{22} + 792 q^{25} - 880 q^{31} - 1792 q^{34} + 1576 q^{37} + 1792 q^{40} - 5768 q^{43} - 160 q^{46} + 2744 q^{49} + 3008 q^{52} + 488 q^{55} + 7552 q^{58} - 2560 q^{61} - 4096 q^{64} + 23784 q^{67} + 1568 q^{70} - 13176 q^{73} - 8960 q^{76} - 9592 q^{79} - 3360 q^{82} - 39880 q^{85} + 8960 q^{88} + 5096 q^{91} + 4608 q^{94} - 14016 q^{97}+O(q^{100}) $ 8 * q - 64 * q^4 - 224 * q^10 - 376 * q^13 + 512 * q^16 + 1120 * q^19 - 1120 * q^22 + 792 * q^25 - 880 * q^31 - 1792 * q^34 + 1576 * q^37 + 1792 * q^40 - 5768 * q^43 - 160 * q^46 + 2744 * q^49 + 3008 * q^52 + 488 * q^55 + 7552 * q^58 - 2560 * q^61 - 4096 * q^64 + 23784 * q^67 + 1568 * q^70 - 13176 * q^73 - 8960 * q^76 - 9592 * q^79 - 3360 * q^82 - 39880 * q^85 + 8960 * q^88 + 5096 * q^91 + 4608 * q^94 - 14016 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.82843i	0.707107i
$2$	2.82843i	0.707107i
$3$	0	0
$3$	0	0
$4$	−8.00000	−0.500000
$5$	3.12468i	0.124987i	0.998045	+	0.0624936i	$0.0199053\pi$
$5$	3.12468i	0.124987i	−0.998045	+	0.0624936i	$0.980095\pi$
$6$	0	0
$7$	18.5203	0.377964
$7$	18.5203	0.377964
$8$	− 22.6274i	− 0.353553i
$9$	0	0
$10$	−8.83793	−0.0883793
$11$	− 141.266i	− 1.16749i	−0.811938	−	0.583744i	$-0.801587\pi$
$11$	− 141.266i	− 1.16749i	0.811938	−	0.583744i	$-0.198413\pi$
$12$	0	0
$13$	−237.623	−1.40606	−0.703028	−	0.711162i	$-0.748169\pi$
$13$	−237.623	−1.40606	−0.703028	+	0.711162i	$0.748169\pi$
$14$	52.3832i	0.267261i
$15$	0	0
$16$	64.0000	0.250000
$17$	394.261i	1.36423i	0.731247	+	0.682113i	$0.238939\pi$
$17$	394.261i	1.36423i	−0.731247	+	0.682113i	$0.761061\pi$
$18$	0	0
$19$	535.891	1.48446	0.742232	−	0.670143i	$-0.233767\pi$
$19$	535.891	1.48446	0.742232	+	0.670143i	$0.233767\pi$
$20$	− 24.9974i	− 0.0624936i
$21$	0	0
$22$	399.561	0.825539
$23$	487.858i	0.922227i	0.887341	+	0.461114i	$0.152550\pi$
$23$	487.858i	0.922227i	−0.887341	+	0.461114i	$0.847450\pi$
$24$	0	0
$25$	615.236	0.984378
$26$	− 672.101i	− 0.994232i
$27$	0	0
$28$	−148.162	−0.188982
$29$	− 1201.77i	− 1.42897i	−0.699649	−	0.714487i	$-0.746660\pi$
$29$	− 1201.77i	− 1.42897i	0.699649	−	0.714487i	$-0.253340\pi$
$30$	0	0
$31$	296.662	0.308701	0.154350	−	0.988016i	$-0.450672\pi$
$31$	296.662	0.308701	0.154350	+	0.988016i	$0.450672\pi$
$32$	181.019i	0.176777i
$33$	0	0
$34$	−1115.14	−0.964653
$35$	57.8699i	0.0472407i
$36$	0	0
$37$	−380.977	−0.278289	−0.139144	−	0.990272i	$-0.544435\pi$
$37$	−380.977	−0.278289	−0.139144	+	0.990272i	$0.544435\pi$
$38$	1515.73i	1.04967i
$39$	0	0
$40$	70.7034	0.0441896
$41$	− 73.6638i	− 0.0438214i	−0.999760	−	0.0219107i	$-0.993025\pi$
$41$	− 73.6638i	− 0.0438214i	0.999760	−	0.0219107i	$-0.00697495\pi$
$42$	0	0
$43$	2912.26	1.57504	0.787522	−	0.616287i	$-0.211364\pi$
$43$	2912.26	1.57504	0.787522	+	0.616287i	$0.211364\pi$
$44$	1130.13i	0.583744i
$45$	0	0
$46$	−1379.87	−0.652113
$47$	2845.13i	1.28797i	0.765037	+	0.643986i	$0.222721\pi$
$47$	2845.13i	1.28797i	−0.765037	+	0.643986i	$0.777279\pi$
$48$	0	0
$49$	343.000	0.142857
$50$	1740.15i	0.696061i
$51$	0	0
$52$	1900.99	0.703028
$53$	2125.45i	0.756657i	0.925671	+	0.378328i	$0.123501\pi$
$53$	2125.45i	0.756657i	−0.925671	+	0.378328i	$0.876499\pi$
$54$	0	0
$55$	441.411	0.145921
$56$	− 419.066i	− 0.133631i
$57$	0	0
$58$	3399.11	1.01044
$59$	5486.16i	1.57603i	0.615656	+	0.788015i	$0.288891\pi$
$59$	5486.16i	1.57603i	−0.615656	+	0.788015i	$0.711109\pi$
$60$	0	0
$61$	−5032.38	−1.35243	−0.676214	−	0.736705i	$-0.736381\pi$
$61$	−5032.38	−1.35243	−0.676214	+	0.736705i	$0.736381\pi$
$62$	839.086i	0.218284i
$63$	0	0
$64$	−512.000	−0.125000
$65$	− 742.497i	− 0.175739i
$66$	0	0
$67$	7342.60	1.63569	0.817844	−	0.575440i	$-0.195169\pi$
$67$	7342.60	1.63569	0.817844	+	0.575440i	$0.195169\pi$
$68$	− 3154.09i	− 0.682113i
$69$	0	0
$70$	−163.681	−0.0334042
$71$	8600.64i	1.70614i	0.521799	+	0.853068i	$0.325261\pi$
$71$	8600.64i	1.70614i	−0.521799	+	0.853068i	$0.674739\pi$
$72$	0	0
$73$	2165.28	0.406320	0.203160	−	0.979146i	$-0.434879\pi$
$73$	2165.28	0.406320	0.203160	+	0.979146i	$0.434879\pi$
$74$	− 1077.57i	− 0.196780i
$75$	0	0
$76$	−4287.13	−0.742232
$77$	− 2616.28i	− 0.441269i
$78$	0	0
$79$	−5796.36	−0.928756	−0.464378	−	0.885637i	$-0.653722\pi$
$79$	−5796.36	−0.928756	−0.464378	+	0.885637i	$0.653722\pi$
$80$	199.979i	0.0312468i
$81$	0	0
$82$	208.353	0.0309864
$83$	2410.24i	0.349868i	0.984580	+	0.174934i	$0.0559713\pi$
$83$	2410.24i	0.349868i	−0.984580	+	0.174934i	$0.944029\pi$
$84$	0	0
$85$	−1231.94	−0.170511
$86$	8237.10i	1.11372i
$87$	0	0
$88$	−3196.49	−0.412769
$89$	13791.5i	1.74113i	0.492055	+	0.870564i	$0.336246\pi$
$89$	13791.5i	1.74113i	−0.492055	+	0.870564i	$0.663754\pi$
$90$	0	0
$91$	−4400.85	−0.531439
$92$	− 3902.87i	− 0.461114i
$93$	0	0
$94$	−8047.25	−0.910734
$95$	1674.49i	0.185539i
$96$	0	0
$97$	10613.0	1.12797	0.563984	−	0.825786i	$-0.309268\pi$
$97$	10613.0	1.12797	0.563984	+	0.825786i	$0.309268\pi$
$98$	970.151i	0.101015i
$99$	0	0
$100$	−4921.89	−0.492189
$101$	− 4424.36i	− 0.433718i	−0.976203	−	0.216859i	$-0.930419\pi$
$101$	− 4424.36i	− 0.433718i	0.976203	−	0.216859i	$-0.0695812\pi$
$102$	0	0
$103$	11498.7	1.08387	0.541933	−	0.840422i	$-0.317693\pi$
$103$	11498.7	1.08387	0.541933	+	0.840422i	$0.317693\pi$
$104$	5376.81i	0.497116i
$105$	0	0
$106$	−6011.68	−0.535037
$107$	− 13254.7i	− 1.15771i	−0.815429	−	0.578857i	$-0.803499\pi$
$107$	− 13254.7i	− 1.15771i	0.815429	−	0.578857i	$-0.196501\pi$
$108$	0	0
$109$	7021.44	0.590981	0.295490	−	0.955346i	$-0.404517\pi$
$109$	7021.44	0.590981	0.295490	+	0.955346i	$0.404517\pi$
$110$	1248.50i	0.103182i
$111$	0	0
$112$	1185.30	0.0944911
$113$	16163.8i	1.26586i	0.774208	+	0.632932i	$0.218149\pi$
$113$	16163.8i	1.26586i	−0.774208	+	0.632932i	$0.781851\pi$
$114$	0	0
$115$	−1524.40	−0.115267
$116$	9614.14i	0.714487i
$117$	0	0
$118$	−15517.2	−1.11442
$119$	7301.82i	0.515629i
$120$	0	0
$121$	−5315.10	−0.363028
$122$	− 14233.7i	− 0.956311i
$123$	0	0
$124$	−2373.29	−0.154350
$125$	3875.34i	0.248022i
$126$	0	0
$127$	−29047.0	−1.80092	−0.900458	−	0.434942i	$-0.856769\pi$
$127$	−29047.0	−1.80092	−0.900458	+	0.434942i	$0.856769\pi$
$128$	− 1448.15i	− 0.0883883i
$129$	0	0
$130$	2100.10	0.124266
$131$	− 33352.2i	− 1.94349i	−0.236032	−	0.971745i	$-0.575847\pi$
$131$	− 33352.2i	− 1.94349i	0.236032	−	0.971745i	$-0.424153\pi$
$132$	0	0
$133$	9924.85	0.561075
$134$	20768.0i	1.15661i
$135$	0	0
$136$	8921.11	0.482326
$137$	3627.12i	0.193251i	0.995321	+	0.0966254i	$0.0308049\pi$
$137$	3627.12i	0.193251i	−0.995321	+	0.0966254i	$0.969195\pi$
$138$	0	0
$139$	9107.22	0.471364	0.235682	−	0.971830i	$-0.424268\pi$
$139$	9107.22	0.471364	0.235682	+	0.971830i	$0.424268\pi$
$140$	− 462.959i	− 0.0236204i
$141$	0	0
$142$	−24326.3	−1.20642
$143$	33568.1i	1.64155i
$144$	0	0
$145$	3755.14	0.178603
$146$	6124.33i	0.287312i
$147$	0	0
$148$	3047.82	0.139144
$149$	9668.97i	0.435520i	0.976002	+	0.217760i	$0.0698749\pi$
$149$	9668.97i	0.435520i	−0.976002	+	0.217760i	$0.930125\pi$
$150$	0	0
$151$	−21411.0	−0.939039	−0.469520	−	0.882922i	$-0.655573\pi$
$151$	−21411.0	−0.939039	−0.469520	+	0.882922i	$0.655573\pi$
$152$	− 12125.8i	− 0.524837i
$153$	0	0
$154$	7399.97	0.312024
$155$	926.972i	0.0385836i
$156$	0	0
$157$	26796.2	1.08711	0.543555	−	0.839374i	$-0.317078\pi$
$157$	26796.2	1.08711	0.543555	+	0.839374i	$0.317078\pi$
$158$	− 16394.6i	− 0.656729i
$159$	0	0
$160$	−565.627	−0.0220948
$161$	9035.26i	0.348569i
$162$	0	0
$163$	32250.2	1.21383	0.606914	−	0.794768i	$-0.292407\pi$
$163$	32250.2	1.21383	0.606914	+	0.794768i	$0.292407\pi$
$164$	589.310i	0.0219107i
$165$	0	0
$166$	−6817.20	−0.247394
$167$	− 6114.20i	− 0.219233i	−0.993974	−	0.109617i	$-0.965038\pi$
$167$	− 6114.20i	− 0.219233i	0.993974	−	0.109617i	$-0.0349623\pi$
$168$	0	0
$169$	27903.9	0.976994
$170$	− 3484.45i	− 0.120569i
$171$	0	0
$172$	−23298.0	−0.787522
$173$	− 29551.1i	− 0.987372i	−0.869640	−	0.493686i	$-0.835649\pi$
$173$	− 29551.1i	− 0.987372i	0.869640	−	0.493686i	$-0.164351\pi$
$174$	0	0
$175$	11394.3	0.372060
$176$	− 9041.03i	− 0.291872i
$177$	0	0
$178$	−39008.2	−1.23116
$179$	3311.38i	0.103348i	0.998664	+	0.0516741i	$0.0164557\pi$
$179$	3311.38i	0.103348i	−0.998664	+	0.0516741i	$0.983544\pi$
$180$	0	0
$181$	−38821.0	−1.18497	−0.592487	−	0.805580i	$-0.701854\pi$
$181$	−38821.0	−1.18497	−0.592487	+	0.805580i	$0.701854\pi$
$182$	− 12447.5i	− 0.375784i
$183$	0	0
$184$	11039.0	0.326057
$185$	− 1190.43i	− 0.0347825i
$186$	0	0
$187$	55695.7	1.59272
$188$	− 22761.1i	− 0.643986i
$189$	0	0
$190$	−4736.17	−0.131196
$191$	− 40870.6i	− 1.12033i	−0.828382	−	0.560163i	$-0.810738\pi$
$191$	− 40870.6i	− 1.12033i	0.828382	−	0.560163i	$-0.189262\pi$
$192$	0	0
$193$	38211.7	1.02585	0.512923	−	0.858435i	$-0.328563\pi$
$193$	38211.7	1.02585	0.512923	+	0.858435i	$0.328563\pi$
$194$	30018.2i	0.797593i
$195$	0	0
$196$	−2744.00	−0.0714286
$197$	− 21225.2i	− 0.546915i	−0.961884	−	0.273457i	$-0.911833\pi$
$197$	− 21225.2i	− 0.546915i	0.961884	−	0.273457i	$-0.0881673\pi$
$198$	0	0
$199$	44448.3	1.12240	0.561202	−	0.827679i	$-0.310339\pi$
$199$	44448.3	1.12240	0.561202	+	0.827679i	$0.310339\pi$
$200$	− 13921.2i	− 0.348030i
$201$	0	0
$202$	12514.0	0.306685
$203$	− 22257.0i	− 0.540102i
$204$	0	0
$205$	230.176	0.00547712
$206$	32523.3i	0.766409i
$207$	0	0
$208$	−15207.9	−0.351514
$209$	− 75703.3i	− 1.73309i
$210$	0	0
$211$	902.367	0.0202683	0.0101342	−	0.999949i	$-0.496774\pi$
$211$	902.367	0.0202683	0.0101342	+	0.999949i	$0.496774\pi$
$212$	− 17003.6i	− 0.378328i
$213$	0	0
$214$	37489.9	0.818628
$215$	9099.87i	0.196860i
$216$	0	0
$217$	5494.25	0.116678
$218$	19859.6i	0.417886i
$219$	0	0
$220$	−3531.29	−0.0729605
$221$	− 93685.7i	− 1.91818i
$222$	0	0
$223$	2439.54	0.0490566	0.0245283	−	0.999699i	$-0.492192\pi$
$223$	2439.54	0.0490566	0.0245283	+	0.999699i	$0.492192\pi$
$224$	3352.53i	0.0668153i
$225$	0	0
$226$	−45718.2	−0.895101
$227$	− 3408.89i	− 0.0661548i	−0.999453	−	0.0330774i	$-0.989469\pi$
$227$	− 3408.89i	− 0.0661548i	0.999453	−	0.0330774i	$-0.0105308\pi$
$228$	0	0
$229$	−10788.1	−0.205719	−0.102860	−	0.994696i	$-0.532799\pi$
$229$	−10788.1	−0.205719	−0.102860	+	0.994696i	$0.532799\pi$
$230$	− 4311.66i	− 0.0815058i
$231$	0	0
$232$	−27192.9	−0.505219
$233$	63204.8i	1.16423i	0.813107	+	0.582114i	$0.197774\pi$
$233$	63204.8i	1.16423i	−0.813107	+	0.582114i	$0.802226\pi$
$234$	0	0
$235$	−8890.12	−0.160980
$236$	− 43889.3i	− 0.788015i
$237$	0	0
$238$	−20652.7	−0.364604
$239$	− 42989.7i	− 0.752607i	−0.926496	−	0.376304i	$-0.877195\pi$
$239$	− 42989.7i	− 0.752607i	0.926496	−	0.376304i	$-0.122805\pi$
$240$	0	0
$241$	37709.8	0.649262	0.324631	−	0.945841i	$-0.394760\pi$
$241$	37709.8	0.649262	0.324631	+	0.945841i	$0.394760\pi$
$242$	− 15033.4i	− 0.256700i
$243$	0	0
$244$	40259.1	0.676214
$245$	1071.76i	0.0178553i
$246$	0	0
$247$	−127340.	−2.08724
$248$	− 6712.68i	− 0.109142i
$249$	0	0
$250$	−10961.1	−0.175378
$251$	113426.i	1.80038i	0.435497	+	0.900190i	$0.356573\pi$
$251$	113426.i	1.80038i	−0.435497	+	0.900190i	$0.643427\pi$
$252$	0	0
$253$	68917.8	1.07669
$254$	− 82157.3i	− 1.27344i
$255$	0	0
$256$	4096.00	0.0625000
$257$	− 8447.17i	− 0.127892i	−0.997953	−	0.0639462i	$-0.979631\pi$
$257$	− 8447.17i	− 0.127892i	0.997953	−	0.0639462i	$-0.0203686\pi$
$258$	0	0
$259$	−7055.80	−0.105183
$260$	5939.98i	0.0878695i
$261$	0	0
$262$	94334.4	1.37426
$263$	− 115681.i	− 1.67244i	−0.548391	−	0.836222i	$-0.684760\pi$
$263$	− 115681.i	− 1.67244i	0.548391	−	0.836222i	$-0.315240\pi$
$264$	0	0
$265$	−6641.35	−0.0945724
$266$	28071.7i	0.396740i
$267$	0	0
$268$	−58740.8	−0.817844
$269$	− 34096.2i	− 0.471195i	−0.971851	−	0.235598i	$-0.924295\pi$
$269$	− 34096.2i	− 0.471195i	0.971851	−	0.235598i	$-0.0757048\pi$
$270$	0	0
$271$	68752.9	0.936165	0.468083	−	0.883685i	$-0.344945\pi$
$271$	68752.9	0.936165	0.468083	+	0.883685i	$0.344945\pi$
$272$	25232.7i	0.341056i
$273$	0	0
$274$	−10259.1	−0.136649
$275$	− 86912.0i	− 1.14925i
$276$	0	0
$277$	115701.	1.50792	0.753958	−	0.656923i	$-0.228142\pi$
$277$	115701.	1.50792	0.753958	+	0.656923i	$0.228142\pi$
$278$	25759.1i	0.333304i
$279$	0	0
$280$	1309.45	0.0167021
$281$	9710.36i	0.122977i	0.998108	+	0.0614883i	$0.0195847\pi$
$281$	9710.36i	0.122977i	−0.998108	+	0.0614883i	$0.980415\pi$
$282$	0	0
$283$	16971.5	0.211907	0.105954	−	0.994371i	$-0.466210\pi$
$283$	16971.5	0.211907	0.105954	+	0.994371i	$0.466210\pi$
$284$	− 68805.1i	− 0.853068i
$285$	0	0
$286$	−94945.0	−1.16075
$287$	− 1364.27i	− 0.0165629i
$288$	0	0
$289$	−71920.8	−0.861110
$290$	10621.1i	0.126292i
$291$	0	0
$292$	−17322.2	−0.203160
$293$	46472.1i	0.541324i	0.962674	+	0.270662i	$0.0872425\pi$
$293$	46472.1i	0.541324i	−0.962674	+	0.270662i	$0.912757\pi$
$294$	0	0
$295$	−17142.5	−0.196983
$296$	8620.53i	0.0983899i
$297$	0	0
$298$	−27348.0	−0.307959
$299$	− 115927.i	− 1.29670i
$300$	0	0
$301$	53935.7	0.595311
$302$	− 60559.5i	− 0.664001i
$303$	0	0
$304$	34297.0	0.371116
$305$	− 15724.6i	− 0.169036i
$306$	0	0
$307$	−104419.	−1.10790	−0.553951	−	0.832549i	$-0.686880\pi$
$307$	−104419.	−1.10790	−0.553951	+	0.832549i	$0.686880\pi$
$308$	20930.3i	0.220635i
$309$	0	0
$310$	−2621.87	−0.0272828
$311$	114779.i	1.18671i	0.804943	+	0.593353i	$0.202196\pi$
$311$	114779.i	1.18671i	−0.804943	+	0.593353i	$0.797804\pi$
$312$	0	0
$313$	−7552.66	−0.0770924	−0.0385462	−	0.999257i	$-0.512273\pi$
$313$	−7552.66	−0.0770924	−0.0385462	+	0.999257i	$0.512273\pi$
$314$	75791.0i	0.768702i
$315$	0	0
$316$	46370.9	0.464378
$317$	95323.4i	0.948595i	0.880365	+	0.474298i	$0.157298\pi$
$317$	95323.4i	0.948595i	−0.880365	+	0.474298i	$0.842702\pi$
$318$	0	0
$319$	−169769.	−1.66831
$320$	− 1599.84i	− 0.0156234i
$321$	0	0
$322$	−25555.6	−0.246476
$323$	211281.i	2.02514i
$324$	0	0
$325$	−146195.	−1.38409
$326$	91217.3i	0.858306i
$327$	0	0
$328$	−1666.82	−0.0154932
$329$	52692.6i	0.486808i
$330$	0	0
$331$	−66158.3	−0.603849	−0.301924	−	0.953332i	$-0.597629\pi$
$331$	−66158.3	−0.603849	−0.301924	+	0.953332i	$0.597629\pi$
$332$	− 19281.9i	− 0.174934i
$333$	0	0
$334$	17293.6	0.155021
$335$	22943.3i	0.204440i
$336$	0	0
$337$	−140142.	−1.23398	−0.616990	−	0.786971i	$-0.711648\pi$
$337$	−140142.	−1.23398	−0.616990	+	0.786971i	$0.711648\pi$
$338$	78924.2i	0.690839i
$339$	0	0
$340$	9855.51	0.0852553
$341$	− 41908.2i	− 0.360405i
$342$	0	0
$343$	6352.45	0.0539949
$344$	− 65896.8i	− 0.556862i
$345$	0	0
$346$	83583.0	0.698178
$347$	28633.6i	0.237803i	0.992906	+	0.118901i	$0.0379372\pi$
$347$	28633.6i	0.237803i	−0.992906	+	0.118901i	$0.962063\pi$
$348$	0	0
$349$	82009.8	0.673310	0.336655	−	0.941628i	$-0.390704\pi$
$349$	82009.8	0.673310	0.336655	+	0.941628i	$0.390704\pi$
$350$	32228.1i	0.263086i
$351$	0	0
$352$	25571.9	0.206385
$353$	− 58668.4i	− 0.470820i	−0.971896	−	0.235410i	$-0.924357\pi$
$353$	− 58668.4i	− 0.470820i	0.971896	−	0.235410i	$-0.0756433\pi$
$354$	0	0
$355$	−26874.2	−0.213245
$356$	− 110332.i	− 0.870564i
$357$	0	0
$358$	−9365.99	−0.0730782
$359$	− 37840.3i	− 0.293606i	−0.989166	−	0.146803i	$-0.953102\pi$
$359$	− 37840.3i	− 0.293606i	0.989166	−	0.146803i	$-0.0468984\pi$
$360$	0	0
$361$	156859.	1.20363
$362$	− 109802.i	− 0.837904i
$363$	0	0
$364$	35206.8	0.265720
$365$	6765.80i	0.0507848i
$366$	0	0
$367$	76141.5	0.565313	0.282657	−	0.959221i	$-0.408784\pi$
$367$	76141.5	0.565313	0.282657	+	0.959221i	$0.408784\pi$
$368$	31222.9i	0.230557i
$369$	0	0
$370$	3367.05	0.0245950
$371$	39363.9i	0.285989i
$372$	0	0
$373$	85550.6	0.614901	0.307451	−	0.951564i	$-0.400524\pi$
$373$	85550.6	0.614901	0.307451	+	0.951564i	$0.400524\pi$
$374$	157531.i	1.12622i
$375$	0	0
$376$	64378.0	0.455367
$377$	285568.i	2.00922i
$378$	0	0
$379$	−136931.	−0.953287	−0.476644	−	0.879097i	$-0.658147\pi$
$379$	−136931.	−0.953287	−0.476644	+	0.879097i	$0.658147\pi$
$380$	− 13395.9i	− 0.0927695i
$381$	0	0
$382$	115600.	0.792190
$383$	42414.5i	0.289146i	0.989494	+	0.144573i	$0.0461808\pi$
$383$	42414.5i	0.289146i	−0.989494	+	0.144573i	$0.953819\pi$
$384$	0	0
$385$	8175.05	0.0551530
$386$	108079.i	0.725383i
$387$	0	0
$388$	−84904.4	−0.563984
$389$	− 258455.i	− 1.70799i	−0.520277	−	0.853997i	$-0.674171\pi$
$389$	− 258455.i	− 1.70799i	0.520277	−	0.853997i	$-0.325829\pi$
$390$	0	0
$391$	−192343.	−1.25813
$392$	− 7761.20i	− 0.0505076i
$393$	0	0
$394$	60034.0	0.386727
$395$	− 18111.8i	− 0.116083i
$396$	0	0
$397$	114157.	0.724303	0.362152	−	0.932119i	$-0.382042\pi$
$397$	114157.	0.724303	0.362152	+	0.932119i	$0.382042\pi$
$398$	125719.i	0.793660i
$399$	0	0
$400$	39375.1	0.246095
$401$	− 140116.i	− 0.871362i	−0.900101	−	0.435681i	$-0.856508\pi$
$401$	− 140116.i	− 0.871362i	0.900101	−	0.435681i	$-0.143492\pi$
$402$	0	0
$403$	−70493.8	−0.434051
$404$	35394.9i	0.216859i
$405$	0	0
$406$	62952.4	0.381909
$407$	53819.1i	0.324899i
$408$	0	0
$409$	−165912.	−0.991817	−0.495908	−	0.868375i	$-0.665165\pi$
$409$	−165912.	−0.991817	−0.495908	+	0.868375i	$0.665165\pi$
$410$	651.035i	0.00387291i
$411$	0	0
$412$	−91989.9	−0.541933
$413$	101605.i	0.595683i
$414$	0	0
$415$	−7531.24	−0.0437291
$416$	− 43014.4i	− 0.248558i
$417$	0	0
$418$	214121.	1.22548
$419$	130600.i	0.743903i	0.928252	+	0.371952i	$0.121311\pi$
$419$	130600.i	0.743903i	−0.928252	+	0.371952i	$0.878689\pi$
$420$	0	0
$421$	−62110.0	−0.350427	−0.175213	−	0.984530i	$-0.556062\pi$
$421$	−62110.0	−0.350427	−0.175213	+	0.984530i	$0.556062\pi$
$422$	2552.28i	0.0143319i
$423$	0	0
$424$	48093.4	0.267519
$425$	242564.i	1.34291i
$426$	0	0
$427$	−93201.1	−0.511170
$428$	106037.i	0.578857i
$429$	0	0
$430$	−25738.3	−0.139201
$431$	− 112404.i	− 0.605100i	−0.953133	−	0.302550i	$-0.902162\pi$
$431$	− 112404.i	− 0.605100i	0.953133	−	0.302550i	$-0.0978380\pi$
$432$	0	0
$433$	64860.0	0.345940	0.172970	−	0.984927i	$-0.444664\pi$
$433$	64860.0	0.345940	0.172970	+	0.984927i	$0.444664\pi$
$434$	15540.1i	0.0825038i
$435$	0	0
$436$	−56171.5	−0.295490
$437$	261439.i	1.36901i
$438$	0	0
$439$	146510.	0.760218	0.380109	−	0.924942i	$-0.375887\pi$
$439$	146510.	0.760218	0.380109	+	0.924942i	$0.375887\pi$
$440$	− 9987.99i	− 0.0515909i
$441$	0	0
$442$	264983.	1.35636
$443$	130291.i	0.663908i	0.943296	+	0.331954i	$0.107708\pi$
$443$	130291.i	0.663908i	−0.943296	+	0.331954i	$0.892292\pi$
$444$	0	0
$445$	−43094.0	−0.217619
$446$	6900.05i	0.0346883i
$447$	0	0
$448$	−9482.37	−0.0472456
$449$	− 93245.1i	− 0.462523i	−0.972892	−	0.231262i	$-0.925715\pi$
$449$	− 93245.1i	− 0.462523i	0.972892	−	0.231262i	$-0.0742853\pi$
$450$	0	0
$451$	−10406.2	−0.0511610
$452$	− 129311.i	− 0.632932i
$453$	0	0
$454$	9641.80	0.0467785
$455$	− 13751.2i	− 0.0664231i
$456$	0	0
$457$	−137893.	−0.660252	−0.330126	−	0.943937i	$-0.607091\pi$
$457$	−137893.	−0.660252	−0.330126	+	0.943937i	$0.607091\pi$
$458$	− 30513.4i	− 0.145465i
$459$	0	0
$460$	12195.2	0.0576333
$461$	324498.i	1.52690i	0.645869	+	0.763448i	$0.276495\pi$
$461$	324498.i	1.52690i	−0.645869	+	0.763448i	$0.723505\pi$
$462$	0	0
$463$	−267434.	−1.24754	−0.623770	−	0.781608i	$-0.714400\pi$
$463$	−267434.	−1.24754	−0.623770	+	0.781608i	$0.714400\pi$
$464$	− 76913.1i	− 0.357244i
$465$	0	0
$466$	−178770.	−0.823234
$467$	− 77333.7i	− 0.354597i	−0.984157	−	0.177299i	$-0.943264\pi$
$467$	− 77333.7i	− 0.354597i	0.984157	−	0.177299i	$-0.0567358\pi$
$468$	0	0
$469$	135987.	0.618232
$470$	− 25145.1i	− 0.113830i
$471$	0	0
$472$	124138.	0.557211
$473$	− 411403.i	− 1.83884i
$474$	0	0
$475$	329700.	1.46127
$476$	− 58414.5i	− 0.257814i
$477$	0	0
$478$	121593.	0.532174
$479$	291569.i	1.27078i	0.772191	+	0.635390i	$0.219161\pi$
$479$	291569.i	1.27078i	−0.772191	+	0.635390i	$0.780839\pi$
$480$	0	0
$481$	90529.1	0.391290
$482$	106659.i	0.459098i
$483$	0	0
$484$	42520.8	0.181514
$485$	33162.4i	0.140981i
$486$	0	0
$487$	−98706.0	−0.416184	−0.208092	−	0.978109i	$-0.566725\pi$
$487$	−98706.0	−0.416184	−0.208092	+	0.978109i	$0.566725\pi$
$488$	113870.i	0.478155i
$489$	0	0
$490$	−3031.41	−0.0126256
$491$	− 68699.3i	− 0.284963i	−0.989797	−	0.142482i	$-0.954492\pi$
$491$	− 68699.3i	− 0.284963i	0.989797	−	0.142482i	$-0.0455082\pi$
$492$	0	0
$493$	473810.	1.94944
$494$	− 360173.i	− 1.47590i
$495$	0	0
$496$	18986.3	0.0771752
$497$	159286.i	0.644859i
$498$	0	0
$499$	−180484.	−0.724834	−0.362417	−	0.932016i	$-0.618048\pi$
$499$	−180484.	−0.724834	−0.362417	+	0.932016i	$0.618048\pi$
$500$	− 31002.7i	− 0.124011i
$501$	0	0
$502$	−320817.	−1.27306
$503$	− 170691.i	− 0.674645i	−0.941389	−	0.337323i	$-0.890479\pi$
$503$	− 170691.i	− 0.674645i	0.941389	−	0.337323i	$-0.109521\pi$
$504$	0	0
$505$	13824.7	0.0542092
$506$	194929.i	0.761334i
$507$	0	0
$508$	232376.	0.900458
$509$	87777.9i	0.338805i	0.985547	+	0.169403i	$0.0541838\pi$
$509$	87777.9i	0.338805i	−0.985547	+	0.169403i	$0.945816\pi$
$510$	0	0
$511$	40101.5	0.153574
$512$	11585.2i	0.0441942i
$513$	0	0
$514$	23892.2	0.0904336
$515$	35929.9i	0.135469i
$516$	0	0
$517$	401921.	1.50369
$518$	− 19956.8i	− 0.0743758i
$519$	0	0
$520$	−16800.8	−0.0621331
$521$	433477.i	1.59695i	0.602030	+	0.798473i	$0.294359\pi$
$521$	433477.i	1.59695i	−0.602030	+	0.798473i	$0.705641\pi$
$522$	0	0
$523$	−380650.	−1.39163	−0.695813	−	0.718223i	$-0.744956\pi$
$523$	−380650.	−1.39163	−0.695813	+	0.718223i	$0.744956\pi$
$524$	266818.i	0.971745i
$525$	0	0
$526$	327196.	1.18260
$527$	116962.i	0.421137i
$528$	0	0
$529$	41835.4	0.149497
$530$	− 18784.6i	− 0.0668728i
$531$	0	0
$532$	−79398.8	−0.280537
$533$	17504.3i	0.0616154i
$534$	0	0
$535$	41416.6	0.144699
$536$	− 166144.i	− 0.578303i
$537$	0	0
$538$	96438.5	0.333185
$539$	− 48454.3i	− 0.166784i
$540$	0	0
$541$	−158944.	−0.543061	−0.271531	−	0.962430i	$-0.587530\pi$
$541$	−158944.	−0.543061	−0.271531	+	0.962430i	$0.587530\pi$
$542$	194463.i	0.661969i
$543$	0	0
$544$	−71368.9	−0.241163
$545$	21939.8i	0.0738650i
$546$	0	0
$547$	−500492.	−1.67272	−0.836358	−	0.548183i	$-0.815320\pi$
$547$	−500492.	−1.67272	−0.836358	+	0.548183i	$0.815320\pi$
$548$	− 29017.0i	− 0.0966254i
$549$	0	0
$550$	245824.	0.812642
$551$	− 644017.i	− 2.12126i
$552$	0	0
$553$	−107350.	−0.351037
$554$	327251.i	1.06626i
$555$	0	0
$556$	−72857.7	−0.235682
$557$	− 411990.i	− 1.32793i	−0.747762	−	0.663967i	$-0.768872\pi$
$557$	− 411990.i	− 1.32793i	0.747762	−	0.663967i	$-0.231128\pi$
$558$	0	0
$559$	−692020.	−2.21460
$560$	3703.67i	0.0118102i
$561$	0	0
$562$	−27465.0	−0.0869576
$563$	237439.i	0.749091i	0.927209	+	0.374545i	$0.122201\pi$
$563$	237439.i	0.749091i	−0.927209	+	0.374545i	$0.877799\pi$
$564$	0	0
$565$	−50506.7	−0.158217
$566$	48002.5i	0.149841i
$567$	0	0
$568$	194610.	0.603210
$569$	− 548677.i	− 1.69470i	−0.531037	−	0.847349i	$-0.678197\pi$
$569$	− 548677.i	− 1.69470i	0.531037	−	0.847349i	$-0.321803\pi$
$570$	0	0
$571$	205016.	0.628806	0.314403	−	0.949290i	$-0.398196\pi$
$571$	205016.	0.628806	0.314403	+	0.949290i	$0.398196\pi$
$572$	− 268545.i	− 0.820777i
$573$	0	0
$574$	3858.75	0.0117118
$575$	300148.i	0.907820i
$576$	0	0
$577$	−292763.	−0.879356	−0.439678	−	0.898155i	$-0.644907\pi$
$577$	−292763.	−0.879356	−0.439678	+	0.898155i	$0.644907\pi$
$578$	− 203423.i	− 0.608897i
$579$	0	0
$580$	−30041.1	−0.0893017
$581$	44638.3i	0.132238i
$582$	0	0
$583$	300254.	0.883388
$584$	− 48994.7i	− 0.143656i
$585$	0	0
$586$	−131443.	−0.382774
$587$	− 75489.1i	− 0.219083i	−0.993982	−	0.109541i	$-0.965062\pi$
$587$	− 75489.1i	− 0.219083i	0.993982	−	0.109541i	$-0.0349382\pi$
$588$	0	0
$589$	158978.	0.458255
$590$	− 48486.3i	− 0.139288i
$591$	0	0
$592$	−24382.5	−0.0695722
$593$	− 664448.i	− 1.88952i	−0.327764	−	0.944760i	$-0.606295\pi$
$593$	− 664448.i	− 1.88952i	0.327764	−	0.944760i	$-0.393705\pi$
$594$	0	0
$595$	−22815.8	−0.0644470
$596$	− 77351.8i	− 0.217760i
$597$	0	0
$598$	327890.	0.916908
$599$	− 521123.i	− 1.45240i	−0.687483	−	0.726201i	$-0.741284\pi$
$599$	− 521123.i	− 1.45240i	0.687483	−	0.726201i	$-0.258716\pi$
$600$	0	0
$601$	52148.0	0.144374	0.0721869	−	0.997391i	$-0.477002\pi$
$601$	52148.0	0.144374	0.0721869	+	0.997391i	$0.477002\pi$
$602$	152553.i	0.420948i
$603$	0	0
$604$	171288.	0.469520
$605$	− 16608.0i	− 0.0453739i
$606$	0	0
$607$	−342719.	−0.930168	−0.465084	−	0.885267i	$-0.653976\pi$
$607$	−342719.	−0.930168	−0.465084	+	0.885267i	$0.653976\pi$
$608$	97006.7i	0.262419i
$609$	0	0
$610$	44475.8	0.119527
$611$	− 676070.i	− 1.81096i
$612$	0	0
$613$	−702183.	−1.86866	−0.934328	−	0.356415i	$-0.883999\pi$
$613$	−702183.	−1.86866	−0.934328	+	0.356415i	$0.883999\pi$
$614$	− 295340.i	− 0.783405i
$615$	0	0
$616$	−59199.7	−0.156012
$617$	5722.23i	0.0150312i	0.999972	+	0.00751562i	$0.00239232\pi$
$617$	5722.23i	0.0150312i	−0.999972	+	0.00751562i	$0.997608\pi$
$618$	0	0
$619$	−493596.	−1.28822	−0.644110	−	0.764933i	$-0.722772\pi$
$619$	−493596.	−1.28822	−0.644110	+	0.764933i	$0.722772\pi$
$620$	− 7415.78i	− 0.0192918i
$621$	0	0
$622$	−324645.	−0.839127
$623$	255422.i	0.658085i
$624$	0	0
$625$	372414.	0.953379
$626$	− 21362.2i	− 0.0545126i
$627$	0	0
$628$	−214369.	−0.543555
$629$	− 150204.i	− 0.379648i
$630$	0	0
$631$	369529.	0.928089	0.464045	−	0.885812i	$-0.346398\pi$
$631$	369529.	0.928089	0.464045	+	0.885812i	$0.346398\pi$
$632$	131157.i	0.328365i
$633$	0	0
$634$	−269615.	−0.670758
$635$	− 90762.5i	− 0.225091i
$636$	0	0
$637$	−81504.9	−0.200865
$638$	− 480179.i	− 1.17967i
$639$	0	0
$640$	4525.02	0.0110474
$641$	685619.i	1.66866i	0.551269	+	0.834328i	$0.314144\pi$
$641$	685619.i	1.66866i	−0.551269	+	0.834328i	$0.685856\pi$
$642$	0	0
$643$	90659.6	0.219276	0.109638	−	0.993972i	$-0.465031\pi$
$643$	90659.6	0.219276	0.109638	+	0.993972i	$0.465031\pi$
$644$	− 72282.1i	− 0.174285i
$645$	0	0
$646$	−597593.	−1.43199
$647$	− 758270.i	− 1.81140i	−0.423916	−	0.905702i	$-0.639345\pi$
$647$	− 758270.i	− 1.81140i	0.423916	−	0.905702i	$-0.360655\pi$
$648$	0	0
$649$	775008.	1.84000
$650$	− 413501.i	− 0.978700i
$651$	0	0
$652$	−258001.	−0.606914
$653$	261552.i	0.613383i	0.951809	+	0.306692i	$0.0992220\pi$
$653$	261552.i	0.613383i	−0.951809	+	0.306692i	$0.900778\pi$
$654$	0	0
$655$	104215.	0.242911
$656$	− 4714.48i	− 0.0109554i
$657$	0	0
$658$	−149037.	−0.344225
$659$	517272.i	1.19110i	0.803319	+	0.595549i	$0.203066\pi$
$659$	517272.i	1.19110i	−0.803319	+	0.595549i	$0.796934\pi$
$660$	0	0
$661$	−626741.	−1.43445	−0.717224	−	0.696842i	$-0.754588\pi$
$661$	−626741.	−1.43445	−0.717224	+	0.696842i	$0.754588\pi$
$662$	− 187124.i	− 0.426985i
$663$	0	0
$664$	54537.6	0.123697
$665$	31012.0i	0.0701271i
$666$	0	0
$667$	586292.	1.31784
$668$	48913.6i	0.109617i
$669$	0	0
$670$	−64893.4	−0.144561
$671$	710905.i	1.57894i
$672$	0	0
$673$	742305.	1.63890	0.819450	−	0.573151i	$-0.194279\pi$
$673$	742305.	1.63890	0.819450	+	0.573151i	$0.194279\pi$
$674$	− 396381.i	− 0.872555i
$675$	0	0
$676$	−223231.	−0.488497
$677$	− 604523.i	− 1.31897i	−0.751717	−	0.659486i	$-0.770774\pi$
$677$	− 604523.i	− 1.31897i	0.751717	−	0.659486i	$-0.229226\pi$
$678$	0	0
$679$	196556.	0.426332
$680$	27875.6i	0.0602846i
$681$	0	0
$682$	118534.	0.254845
$683$	− 720939.i	− 1.54546i	−0.634737	−	0.772728i	$-0.718891\pi$
$683$	− 720939.i	− 1.54546i	0.634737	−	0.772728i	$-0.281109\pi$
$684$	0	0
$685$	−11333.6	−0.0241539
$686$	17967.4i	0.0381802i
$687$	0	0
$688$	186384.	0.393761
$689$	− 505057.i	− 1.06390i
$690$	0	0
$691$	−588601.	−1.23272	−0.616360	−	0.787464i	$-0.711393\pi$
$691$	−588601.	−1.23272	−0.616360	+	0.787464i	$0.711393\pi$
$692$	236409.i	0.493686i
$693$	0	0
$694$	−80988.0	−0.168152
$695$	28457.1i	0.0589144i
$696$	0	0
$697$	29042.8	0.0597823
$698$	231959.i	0.476102i
$699$	0	0
$700$	−91154.7	−0.186030
$701$	− 154697.i	− 0.314809i	−0.987534	−	0.157404i	$-0.949687\pi$
$701$	− 154697.i	− 0.314809i	0.987534	−	0.157404i	$-0.0503126\pi$
$702$	0	0
$703$	−204162.	−0.413109
$704$	72328.2i	0.145936i
$705$	0	0
$706$	165939.	0.332920
$707$	− 81940.3i	− 0.163930i
$708$	0	0
$709$	715382.	1.42313	0.711567	−	0.702619i	$-0.247986\pi$
$709$	715382.	1.42313	0.711567	+	0.702619i	$0.247986\pi$
$710$	− 76011.8i	− 0.150787i
$711$	0	0
$712$	312066.	0.615582
$713$	144729.i	0.284692i
$714$	0	0
$715$	−104890.	−0.205173
$716$	− 26491.0i	− 0.0516741i
$717$	0	0
$718$	107028.	0.207611
$719$	371799.i	0.719202i	0.933106	+	0.359601i	$0.117087\pi$
$719$	371799.i	0.719202i	−0.933106	+	0.359601i	$0.882913\pi$
$720$	0	0
$721$	212960.	0.409663
$722$	443663.i	0.851097i
$723$	0	0
$724$	310568.	0.592487
$725$	− 739371.i	− 1.40665i
$726$	0	0
$727$	52138.1	0.0986476	0.0493238	−	0.998783i	$-0.484293\pi$
$727$	52138.1	0.0986476	0.0493238	+	0.998783i	$0.484293\pi$
$728$	99579.8i	0.187892i
$729$	0	0
$730$	−19136.6	−0.0359103
$731$	1.14819e6i	2.14871i
$732$	0	0
$733$	324955.	0.604805	0.302403	−	0.953180i	$-0.402211\pi$
$733$	324955.	0.604805	0.302403	+	0.953180i	$0.402211\pi$
$734$	215361.i	0.399737i
$735$	0	0
$736$	−88311.8	−0.163028
$737$	− 1.03726e6i	− 1.90965i
$738$	0	0
$739$	−971247.	−1.77845	−0.889224	−	0.457473i	$-0.848755\pi$
$739$	−971247.	−1.77845	−0.889224	+	0.457473i	$0.848755\pi$
$740$	9523.45i	0.0173913i
$741$	0	0
$742$	−111338.	−0.202225
$743$	681732.i	1.23491i	0.786605	+	0.617456i	$0.211837\pi$
$743$	681732.i	1.23491i	−0.786605	+	0.617456i	$0.788163\pi$
$744$	0	0
$745$	−30212.4	−0.0544344
$746$	241974.i	0.434801i
$747$	0	0
$748$	−445566.	−0.796358
$749$	− 245480.i	− 0.437575i
$750$	0	0
$751$	−317894.	−0.563641	−0.281821	−	0.959467i	$-0.590938\pi$
$751$	−317894.	−0.563641	−0.281821	+	0.959467i	$0.590938\pi$
$752$	182088.i	0.321993i
$753$	0	0
$754$	−807709.	−1.42073
$755$	− 66902.6i	− 0.117368i
$756$	0	0
$757$	−700233.	−1.22194	−0.610971	−	0.791653i	$-0.709221\pi$
$757$	−700233.	−1.22194	−0.610971	+	0.791653i	$0.709221\pi$
$758$	− 387300.i	− 0.674076i
$759$	0	0
$760$	37889.4	0.0655979
$761$	305818.i	0.528072i	0.964513	+	0.264036i	$0.0850538\pi$
$761$	305818.i	0.528072i	−0.964513	+	0.264036i	$0.914946\pi$
$762$	0	0
$763$	130039.	0.223370
$764$	326965.i	0.560163i
$765$	0	0
$766$	−119966.	−0.204457
$767$	− 1.30364e6i	− 2.21599i
$768$	0	0
$769$	820424.	1.38735	0.693675	−	0.720289i	$-0.255991\pi$
$769$	820424.	1.38735	0.693675	+	0.720289i	$0.255991\pi$
$770$	23122.5i	0.0389990i
$771$	0	0
$772$	−305694.	−0.512923
$773$	483522.i	0.809202i	0.914493	+	0.404601i	$0.132590\pi$
$773$	483522.i	0.809202i	−0.914493	+	0.404601i	$0.867410\pi$
$774$	0	0
$775$	182517.	0.303878
$776$	− 240146.i	− 0.398797i
$777$	0	0
$778$	731022.	1.20773
$779$	− 39475.8i	− 0.0650513i
$780$	0	0
$781$	1.21498e6	1.99189
$782$	− 544030.i	− 0.889629i
$783$	0	0
$784$	21952.0	0.0357143
$785$	83729.4i	0.135875i
$786$	0	0
$787$	−185741.	−0.299888	−0.149944	−	0.988695i	$-0.547909\pi$
$787$	−185741.	−0.299888	−0.149944	+	0.988695i	$0.547909\pi$
$788$	169802.i	0.273457i
$789$	0	0
$790$	51227.8	0.0820827
$791$	299358.i	0.478452i
$792$	0	0
$793$	1.19581e6	1.90159
$794$	322884.i	0.512160i
$795$	0	0
$796$	−355587.	−0.561202
$797$	720505.i	1.13428i	0.823621	+	0.567140i	$0.191950\pi$
$797$	720505.i	1.13428i	−0.823621	+	0.567140i	$0.808050\pi$
$798$	0	0
$799$	−1.12172e6	−1.75708
$800$	111370.i	0.174015i
$801$	0	0
$802$	396307.	0.616146
$803$	− 305880.i	− 0.474374i
$804$	0	0
$805$	−28232.3	−0.0435667
$806$	− 199386.i	− 0.306920i
$807$	0	0
$808$	−100112.	−0.153343
$809$	− 686330.i	− 1.04866i	−0.851514	−	0.524331i	$-0.824315\pi$
$809$	− 686330.i	− 1.04866i	0.851514	−	0.524331i	$-0.175685\pi$
$810$	0	0
$811$	1.24830e6	1.89791	0.948955	−	0.315412i	$-0.102143\pi$
$811$	1.24830e6	1.89791	0.948955	+	0.315412i	$0.102143\pi$
$812$	178056.i	0.270051i
$813$	0	0
$814$	−152224.	−0.229738
$815$	100771.i	0.151713i
$816$	0	0
$817$	1.56065e6	2.33810
$818$	− 469270.i	− 0.701320i
$819$	0	0
$820$	−1841.41	−0.00273856
$821$	842160.i	1.24942i	0.780857	+	0.624710i	$0.214783\pi$
$821$	842160.i	1.24942i	−0.780857	+	0.624710i	$0.785217\pi$
$822$	0	0
$823$	−105260.	−0.155404	−0.0777019	−	0.996977i	$-0.524758\pi$
$823$	−105260.	−0.155404	−0.0777019	+	0.996977i	$0.524758\pi$
$824$	− 260187.i	− 0.383204i
$825$	0	0
$826$	−287383.	−0.421212
$827$	373318.i	0.545844i	0.962036	+	0.272922i	$0.0879900\pi$
$827$	373318.i	0.545844i	−0.962036	+	0.272922i	$0.912010\pi$
$828$	0	0
$829$	867025.	1.26160	0.630801	−	0.775944i	$-0.282726\pi$
$829$	867025.	1.26160	0.630801	+	0.775944i	$0.282726\pi$
$830$	− 21301.6i	− 0.0309211i
$831$	0	0
$832$	121663.	0.175757
$833$	135232.i	0.194889i
$834$	0	0
$835$	19104.9	0.0274014
$836$	605626.i	0.866547i
$837$	0	0
$838$	−369394.	−0.526019
$839$	686493.i	0.975242i	0.873055	+	0.487621i	$0.162135\pi$
$839$	686493.i	0.975242i	−0.873055	+	0.487621i	$0.837865\pi$
$840$	0	0
$841$	−736964.	−1.04197
$842$	− 175674.i	− 0.247789i
$843$	0	0
$844$	−7218.94	−0.0101342
$845$	87190.8i	0.122112i
$846$	0	0
$847$	−98437.0	−0.137212
$848$	136029.i	0.189164i
$849$	0	0
$850$	−686074.	−0.949583
$851$	− 185863.i	− 0.256645i
$852$	0	0
$853$	974427.	1.33922	0.669609	−	0.742714i	$-0.266462\pi$
$853$	974427.	1.33922	0.669609	+	0.742714i	$0.266462\pi$
$854$	− 263612.i	− 0.361452i
$855$	0	0
$856$	−299919.	−0.409314
$857$	275668.i	0.375339i	0.982232	+	0.187670i	$0.0600934\pi$
$857$	275668.i	0.375339i	−0.982232	+	0.187670i	$0.939907\pi$
$858$	0	0
$859$	176529.	0.239237	0.119619	−	0.992820i	$-0.461833\pi$
$859$	176529.	0.239237	0.119619	+	0.992820i	$0.461833\pi$
$860$	− 72798.9i	− 0.0984301i
$861$	0	0
$862$	317927.	0.427871
$863$	856520.i	1.15005i	0.818136	+	0.575024i	$0.195007\pi$
$863$	856520.i	1.15005i	−0.818136	+	0.575024i	$0.804993\pi$
$864$	0	0
$865$	92337.6	0.123409
$866$	183452.i	0.244617i
$867$	0	0
$868$	−43954.0	−0.0583390
$869$	818829.i	1.08431i
$870$	0	0
$871$	−1.74478e6	−2.29987
$872$	− 158877.i	− 0.208943i
$873$	0	0
$874$	−739461.	−0.968038
$875$	71772.3i	0.0937434i
$876$	0	0
$877$	1.00077e6	1.30118	0.650589	−	0.759430i	$-0.274522\pi$
$877$	1.00077e6	1.30118	0.650589	+	0.759430i	$0.274522\pi$
$878$	414393.i	0.537555i
$879$	0	0
$880$	28250.3	0.0364803
$881$	775427.i	0.999055i	0.866298	+	0.499527i	$0.166493\pi$
$881$	775427.i	0.999055i	−0.866298	+	0.499527i	$0.833507\pi$
$882$	0	0
$883$	160327.	0.205630	0.102815	−	0.994701i	$-0.467215\pi$
$883$	160327.	0.205630	0.102815	+	0.994701i	$0.467215\pi$
$884$	749485.i	0.959089i
$885$	0	0
$886$	−368519.	−0.469454
$887$	183914.i	0.233759i	0.993146	+	0.116879i	$0.0372891\pi$
$887$	183914.i	0.233759i	−0.993146	+	0.116879i	$0.962711\pi$
$888$	0	0
$889$	−537958.	−0.680682
$890$	− 121888.i	− 0.153880i
$891$	0	0
$892$	−19516.3	−0.0245283
$893$	1.52468e6i	1.91195i
$894$	0	0
$895$	−10347.0	−0.0129172
$896$	− 26820.2i	− 0.0334077i
$897$	0	0
$898$	263737.	0.327053
$899$	− 356518.i	− 0.441126i
$900$	0	0
$901$	−837982.	−1.03225
$902$	− 29433.2i	− 0.0361763i
$903$	0	0
$904$	365745.	0.447550
$905$	− 121303.i	− 0.148107i
$906$	0	0
$907$	184492.	0.224266	0.112133	−	0.993693i	$-0.464232\pi$
$907$	184492.	0.224266	0.112133	+	0.993693i	$0.464232\pi$
$908$	27271.1i	0.0330774i
$909$	0	0
$910$	38894.4	0.0469682
$911$	− 979147.i	− 1.17981i	−0.807474	−	0.589904i	$-0.799166\pi$
$911$	− 979147.i	− 1.17981i	0.807474	−	0.589904i	$-0.200834\pi$
$912$	0	0
$913$	340486.	0.408467
$914$	− 390020.i	− 0.466868i
$915$	0	0
$916$	86304.9	0.102860
$917$	− 617692.i	− 0.734570i
$918$	0	0
$919$	1.32373e6	1.56736	0.783679	−	0.621166i	$-0.213341\pi$
$919$	1.32373e6	1.56736	0.783679	+	0.621166i	$0.213341\pi$
$920$	34493.2i	0.0407529i
$921$	0	0
$922$	−917818.	−1.07968
$923$	− 2.04371e6i	− 2.39892i
$924$	0	0
$925$	−234391.	−0.273941
$926$	− 756417.i	− 0.882143i
$927$	0	0
$928$	217543.	0.252609
$929$	− 1.15956e6i	− 1.34358i	−0.740743	−	0.671789i	$-0.765526\pi$
$929$	− 1.15956e6i	− 1.34358i	0.740743	−	0.671789i	$-0.234474\pi$
$930$	0	0
$931$	183811.	0.212066
$932$	− 505638.i	− 0.582114i
$933$	0	0
$934$	218733.	0.250738
$935$	174031.i	0.199069i
$936$	0	0
$937$	−1.49483e6	−1.70260	−0.851300	−	0.524679i	$-0.824185\pi$
$937$	−1.49483e6	−1.70260	−0.851300	+	0.524679i	$0.824185\pi$
$938$	384629.i	0.437156i
$939$	0	0
$940$	71121.0	0.0804900
$941$	1.18590e6i	1.33927i	0.742689	+	0.669637i	$0.233550\pi$
$941$	1.18590e6i	1.33927i	−0.742689	+	0.669637i	$0.766450\pi$
$942$	0	0
$943$	35937.5	0.0404133
$944$	351114.i	0.394007i
$945$	0	0
$946$	1.16362e6	1.30026
$947$	865824.i	0.965450i	0.875772	+	0.482725i	$0.160353\pi$
$947$	865824.i	0.965450i	−0.875772	+	0.482725i	$0.839647\pi$
$948$	0	0
$949$	−514521.	−0.571309
$950$	932532.i	1.03328i
$951$	0	0
$952$	165221.	0.182302
$953$	388687.i	0.427971i	0.976837	+	0.213986i	$0.0686446\pi$
$953$	388687.i	0.427971i	−0.976837	+	0.213986i	$0.931355\pi$
$954$	0	0
$955$	127708.	0.140026
$956$	343917.i	0.376304i
$957$	0	0
$958$	−824682.	−0.898577
$959$	67175.3i	0.0730419i
$960$	0	0
$961$	−835513.	−0.904704
$962$	256055.i	0.276683i
$963$	0	0
$964$	−301678.	−0.324631
$965$	119399.i	0.128218i
$966$	0	0
$967$	192495.	0.205858	0.102929	−	0.994689i	$-0.467179\pi$
$967$	192495.	0.205858	0.102929	+	0.994689i	$0.467179\pi$
$968$	120267.i	0.128350i
$969$	0	0
$970$	−93797.3	−0.0996889
$971$	453950.i	0.481470i	0.970591	+	0.240735i	$0.0773885\pi$
$971$	453950.i	0.481470i	−0.970591	+	0.240735i	$0.922612\pi$
$972$	0	0
$973$	168668.	0.178159
$974$	− 279183.i	− 0.294287i
$975$	0	0
$976$	−322073.	−0.338107
$977$	727881.i	0.762555i	0.924461	+	0.381278i	$0.124516\pi$
$977$	727881.i	0.762555i	−0.924461	+	0.381278i	$0.875484\pi$
$978$	0	0
$979$	1.94827e6	2.03275
$980$	− 8574.12i	− 0.00892765i
$981$	0	0
$982$	194311.	0.201500
$983$	662759.i	0.685881i	0.939357	+	0.342941i	$0.111423\pi$
$983$	662759.i	0.685881i	−0.939357	+	0.342941i	$0.888577\pi$
$984$	0	0
$985$	66322.0	0.0683573
$986$	1.34014e6i	1.37846i
$987$	0	0
$988$	1.01872e6	1.04362
$989$	1.42077e6i	1.45255i
$990$	0	0
$991$	−1.16764e6	−1.18894	−0.594472	−	0.804116i	$-0.702639\pi$
$991$	−1.16764e6	−1.18894	−0.594472	+	0.804116i	$0.702639\pi$
$992$	53701.5i	0.0545711i
$993$	0	0
$994$	−450529.	−0.455984
$995$	138887.i	0.140286i
$996$	0	0
$997$	1.12634e6	1.13313	0.566563	−	0.824019i	$-0.308273\pi$
$997$	1.12634e6	1.13313	0.566563	+	0.824019i	$0.308273\pi$
$998$	− 510487.i	− 0.512535i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.5.b.a.323.6	yes	8
3.2	odd	2	inner	378.5.b.a.323.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.5.b.a.323.3	✓	8	3.2	odd	2	inner
378.5.b.a.323.6	yes	8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	378.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.82843i q^{2} -8.00000 q^{4} +3.12468i q^{5} +18.5203 q^{7} -22.6274i q^{8} +O(q^{10})\)	\(q+2.82843i q^{2} -8.00000 q^{4} +3.12468i q^{5} +18.5203 q^{7} -22.6274i q^{8} -8.83793 q^{10} -141.266i q^{11} -237.623 q^{13} +52.3832i q^{14} +64.0000 q^{16} +394.261i q^{17} +535.891 q^{19} -24.9974i q^{20} +399.561 q^{22} +487.858i q^{23} +615.236 q^{25} -672.101i q^{26} -148.162 q^{28} -1201.77i q^{29} +296.662 q^{31} +181.019i q^{32} -1115.14 q^{34} +57.8699i q^{35} -380.977 q^{37} +1515.73i q^{38} +70.7034 q^{40} -73.6638i q^{41} +2912.26 q^{43} +1130.13i q^{44} -1379.87 q^{46} +2845.13i q^{47} +343.000 q^{49} +1740.15i q^{50} +1900.99 q^{52} +2125.45i q^{53} +441.411 q^{55} -419.066i q^{56} +3399.11 q^{58} +5486.16i q^{59} -5032.38 q^{61} +839.086i q^{62} -512.000 q^{64} -742.497i q^{65} +7342.60 q^{67} -3154.09i q^{68} -163.681 q^{70} +8600.64i q^{71} +2165.28 q^{73} -1077.57i q^{74} -4287.13 q^{76} -2616.28i q^{77} -5796.36 q^{79} +199.979i q^{80} +208.353 q^{82} +2410.24i q^{83} -1231.94 q^{85} +8237.10i q^{86} -3196.49 q^{88} +13791.5i q^{89} -4400.85 q^{91} -3902.87i q^{92} -8047.25 q^{94} +1674.49i q^{95} +10613.0 q^{97} +970.151i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 64 q^{4}+O(q^{10}) \) 8 * q - 64 * q^4	\( 8 q - 64 q^{4} - 224 q^{10} - 376 q^{13} + 512 q^{16} + 1120 q^{19} - 1120 q^{22} + 792 q^{25} - 880 q^{31} - 1792 q^{34} + 1576 q^{37} + 1792 q^{40} - 5768 q^{43} - 160 q^{46} + 2744 q^{49} + 3008 q^{52} + 488 q^{55} + 7552 q^{58} - 2560 q^{61} - 4096 q^{64} + 23784 q^{67} + 1568 q^{70} - 13176 q^{73} - 8960 q^{76} - 9592 q^{79} - 3360 q^{82} - 39880 q^{85} + 8960 q^{88} + 5096 q^{91} + 4608 q^{94} - 14016 q^{97}+O(q^{100}) \) 8 * q - 64 * q^4 - 224 * q^10 - 376 * q^13 + 512 * q^16 + 1120 * q^19 - 1120 * q^22 + 792 * q^25 - 880 * q^31 - 1792 * q^34 + 1576 * q^37 + 1792 * q^40 - 5768 * q^43 - 160 * q^46 + 2744 * q^49 + 3008 * q^52 + 488 * q^55 + 7552 * q^58 - 2560 * q^61 - 4096 * q^64 + 23784 * q^67 + 1568 * q^70 - 13176 * q^73 - 8960 * q^76 - 9592 * q^79 - 3360 * q^82 - 39880 * q^85 + 8960 * q^88 + 5096 * q^91 + 4608 * q^94 - 14016 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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